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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Case One: Let’s first determine whether it has an integrating factor that depends on <span class="process-math">\(x\)</span> only.</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_31_1.html ./knowl/eq2_31_2.html">
\begin{equation}
\frac{ \frac{\partial M}{\partial y}-\frac{\partial N}{\partial x} }{N}=\frac{3 x+2y-(2x +y)}{x^2+xy}=\frac{1}{x}.\tag{2.6.9}
\end{equation}
</div>
<p class="continuation">Thus there is an integrating factor <span class="process-math">\(u\)</span> that is a function of <span class="process-math">\(x\)</span> only and it satisfies the differential equation</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_31_1.html ./knowl/eq2_31_2.html">
\begin{equation*}
\frac{\textrm{d} u}{\textrm{d} x}=\frac{u}{x}.
\end{equation*}
</div>
<p class="continuation">Hence</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_31_1.html ./knowl/eq2_31_2.html">
\begin{equation*}
u(x)=x.
\end{equation*}
</div>
<p class="continuation">Multiplying this integrating factor on (<a href="" class="xref" data-knowl="./knowl/eq2_31_1.html" title="Equation 2.6.8">(2.6.8)</a>), we obtain</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_31_1.html ./knowl/eq2_31_2.html">
\begin{equation}
(3 x^2 y+x y^2)+(x^3+x^2 y)y^{\prime}=0.\tag{2.6.10}
\end{equation}
</div>
<p class="continuation">Check (<a href="" class="xref" data-knowl="./knowl/eq2_31_2.html" title="Equation 2.6.10">(2.6.10)</a>), we find that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_31_1.html ./knowl/eq2_31_2.html">
\begin{equation*}
\frac{\partial M(x, y)}{\partial y}=\frac{\partial (3 x^2 y+x y^2)}{\partial y}=3 x^2+2 x y,\quad 
\frac{\partial N(x, y)}{\partial x}=\frac{\partial (x^3+x^2 y)}{\partial x}=3 x^2+2 x y.
\end{equation*}
</div>
<p class="continuation">Thus, the new equation is exact and it’s solution is given implicitly by</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_31_1.html ./knowl/eq2_31_2.html">
\begin{equation*}
x^3 y+\frac{1}{2} x^2 y^2=C.
\end{equation*}
</div>
<span class="incontext"><a href="sec2_6.html#p-56" class="internal">in-context</a></span>
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